Name: Algebraic Operations: Functions
Uploaded: 2019-07-19
Duration: 7 min 10 s
Description: Now that we know what functions are, what can we do with them? Can we do operations with them? Yes! What won't these functions do? Let us learn more here.

Question

Solve the inequality :

x - \sqrt{1 - | x |} < 0

Accepted Answer

Given, $x - \sqrt{1 - |x|} < 0$

$\Rightarrow x < \sqrt{1 - |x|}$

Let, $y_{1} = x$ and $y_{2} = \sqrt{1 - |x|}$

For the domain of $y_{2}$ ,

$1 - |x| \geq 0$

$\Rightarrow |x| - 1 \leq 0$

$\Rightarrow |x| \leq 1$

$\Rightarrow x \in [- 1, 1]$

The basic function for $y_{2} = \sqrt{1 - |x|}$ is $y = \sqrt{x}$ , whose domain is $x \in [0, \infty)$ and range is $y \in [0, \infty)$ .

After replacing $' x'$ by $x - \sqrt{1 - | x |} < 0$ in $y = \sqrt{x}$ , we get, $y = \sqrt{1 + x}$ .

After replacing $' x'$ by $' - x'$ in $y = \sqrt{1 + x},$ we get, $y = \sqrt{1 - x}$ .

Now, replace $' x'$ by $' |x|'$ in $y = \sqrt{1 - x}$ , we get, $y = \sqrt{1 - |x|}$

Now, plot $y_{1} = x$ and $y_{2} = \sqrt{1 - |x|}$ on the same plane

As shown in the above figure, $y_{1} = x$ and $y_{2} = \sqrt{1 - |x|}$ both intersect in the first quadrant. Hence, here both $' x'$ and $\sqrt{1 - |x|}$ are positive.

$x = \sqrt{1 - |x|}$

Squaring both sides,

$\Rightarrow x^{2} = 1 - |x|$

$\Rightarrow x^{2} + x - 1 = 0$

$\Rightarrow x = \frac{- 1 \pm \sqrt{5}}{2}$

$\Rightarrow x = \frac{\sqrt{5} - 1}{2}$ $; x > 0$

From the above figure,

$x < \sqrt{1 - |x|}$ is true for

$x \in (- \infty, \frac{\sqrt{5} - 1}{2})$ but, $x$ must lie in $[- 1, 1]$ . Hence, the final solution for this inequality is

$x \in [- 1, \frac{\sqrt{5} - 1}{2})$ .

Solve the inequality : $x - \sqrt{1 - | x |} < 0$

Important Questions on Functions

Learn and Prepare for any exam you want !

Solve the inequality : x-1-|x|<0

Important Questions on Functions

Learn and Prepare for any exam you want !

Solve the inequality : $x - \sqrt{1 - | x |} < 0$